Abstract. We present a family of scale-free network model consisting of cliques, which is established by a simple recursive algorithm. We investigate the networks both analytically and numerically. The obtained analytical solutions show that the networks follow a power-law degree distribution, with degree exponent continuously tuned between 2 and 3. The exact expression of clustering coefficient is also provided for the networks. Furthermore, the investigation of the average path length reveals that the networks possess small-world feature. Interestingly, we find that a special case of our model can be mapped into the Yule process. PACS

A vast variety of real-life networks display the ubiquitous presence of scale-free phenomenon and small-world effect, both of which play a significant role in the dynamical processes running on networks. Although various dynamical processes have been investigated in scale-free small-world networks, analytical research about random walks on such networks is much less. In this paper, we will study analytically the scaling of the mean first-passage time (MFPT) for random walks on scale-free small-world networks. To this end, we first map the classical Koch fractal to a network, called Koch network. According to this proposed mapping, we present an iterative algorithm for generating the Koch network, based on which we derive closed-form expressions for the relevant topological features, such as degree distribution, clustering coefficient, average path length, and degree correlations. The obtained solutions show that the Koch network exhibits scale-free behavior and small-world effect. Then, we investigate the standard random walks and trapping issue on the Koch network. Through the recurrence relations derived from the structure of the Koch network, we obtain the exact scaling for the MFPT. We show that in the infinite network order limit, the MFPT grows linearly with the number of all nodes in the network. The obtained analytical results are corroborated by direct extensive numerical calculations. In addition, we also determine the scaling efficiency exponents characterizing random walks on the Koch network.

With the help of recursion relations derived from the self-similar structure, we obtain the exact solution of average path length,dt, for Apollonian networks. In contrast to the well-known numerical resultdt ∝ (ln Nt) 3/4 [Phys. Rev. Lett. 94, 018702 (2005)], our rigorous solution shows that the average path length grows logarithmically asdt ∝ ln Nt in the infinite limit of network size Nt. The extensive numerical calculations completely agree with our closed-form solution.PACS numbers: 89.75. Hc, 89.75.Da, 02.10.Ox, One of the most important properties of complex networks is average path length (APL), which is the mean length of the shortest paths between all pairs of vertices (nodes) [1]. Most real networks have been shown to be small-world or ultra small-world networks [2,3,4,5], that is, their APL d behaves a logarithmic or double logarithmic scaling with the network size N :. It has been established that APL is relevant in many fields regarding real-life networks. In the design or interpretation of routes in architectural design, signal integrity in communication networks, the propagation of diseases or beliefs in social networks or of technology in industrial networks, APL is a natural network statistic to compute and interpret. It is strongly believed that many processes such as routing, searching, and spreading become more efficient when APL is smaller. So far, much attention has been paid to the question of APL [8,9,10,11,12,13].Recently, on the basis of the well-known Apollonian packing [14], Andrade et al. introduced Apollonian networks [15] which were also proposed by Doye and Massen in Ref. [16] simultaneously. Apollonian networks belong to a deterministic growing type of networks, which have drawn much attention from the scientific communities and have turned out to be a useful tool [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Many topological properties of Apollonian networks such as degree distribution, clustering coefficient, and correlations have been determined analytically [15,16], and the effects of the Apollonian networks on several dynamical models have been intensively studied, including Ising model and a magnetic model [15,32,33,34]. Despite the importance and usefulness of the quantity APL, there is no analytical calculations for the APL of Apollonian networks.In this report, we derive an exact formula for the average path length characterizing the Apollonian networks. The analytic method is based on the recursive construction and self-similar structure of Apollonian networks. * Electronic address: sgzhou@fudan.edu.cn Our rigorous result shows that APL grows logarithmically with the number of nodes. The obtained analytical solution modifies the previous numerical result in [15], where the authors claimed that the APL of Apollonian networks scales sub-logarithmically with network size. Our analytical technique could provide a paradigm for computing the APL of deterministic networks.The Apollonian network, denoted as A t (t ≥ 0) after t generations, is constructed as follows [15]: Fo...

Generally, the threshold of percolation in complex networks depends on the underlying structural characterization. However, what topological property plays a predominant role is still unknown, despite the speculation of some authors that degree distribution is a key ingredient. The purpose of this paper is to show that power-law degree distribution itself is not sufficient to characterize the threshold of bond percolation in scale-free networks. To achieve this goal, we first propose a family of scale-free networks with the same degree sequence and obtain by analytical or numerical means several topological features of the networks. Then, by making use of the renormalization-group technique we determine the threshold of bond percolation in our networks. We find an existence of nonzero thresholds and demonstrate that these thresholds can be quite different, which implies that power-law degree distribution does not suffice to characterize the percolation threshold in scale-free networks.

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